Well, we have to remember that we're talking about eighth century. Literacy was not considered highly in the Christian kingdoms. It's not possible to just visit the Bureau of Vital Statistics and ask for documents. Names change depending upon which feudal kingdom they were in and what language was spoken. Remember that in Asturias Bable was spoken. Galacians spoke Galego, Vascos spoke Euskara and Catalanes spoke Catalá.
I found the following in a book published in 1827 on page 92.
Apellidos Castellanos por D. José Godoy Alcántara
"Aldrete. Alderedus, obispo cesar augustano del siglo VII, Ilderedus, Aldoretus, Alderetus, Alderoitus, Alerito, Aldroieto, Aldoitro.Una escritura del 922, en el tumbo de Celanova, comienza: Hacia por patronimicos Alderetiz, Aldretiz, Aldetriz, Aldrey, Aldretez
All of the above were variations of the same name. Also notice above that right after Aldrete is Alderedus, a bishop in the seventh century. I believe that is before the Muslim invasion of Spain.
In some book I have read the conspiracy against Ramiro I way by a Conde de Aldroito. In some places it is written Aldroito (Alderedo).
So, which namevariation do we follow?
As I recollect we are speaking of a rich man. He probably had more than one home. If you have seen pictures it is a small torre that looks more like a fortified house.
Since I have an Alderedus in the seventh century perhaps I'll follow that line for a while to see where it takes me.